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1.
In this paper, we obtain the local well-posedness for the 3D incompressible Hall-magnetohydrodynamics (Hall-MHD) system with \(\varLambda^{2\alpha }u\) and \(\varLambda^{2\beta }B\), \(0<\alpha \le 1\), \(\frac{1}{2}<\beta \le 1\). Our results improve regularity conditions on the initial data of previous works.  相似文献   

2.
Let \(\alpha \) and \(\beta \) be irrational real numbers and \(0<\varepsilon <1/30\). We prove a precise estimate for the number of positive integers \(q\le Q\) that satisfy \(\Vert q\alpha \Vert \cdot \Vert q\beta \Vert <\varepsilon \). If we choose \(\varepsilon \) as a function of Q, we get asymptotics as Q gets large, provided \(\varepsilon Q\) grows quickly enough in terms of the (multiplicative) Diophantine type of \((\alpha ,\beta )\), e.g., if \((\alpha ,\beta )\) is a counterexample to Littlewood’s conjecture, then we only need that \(\varepsilon Q\) tends to infinity. Our result yields a new upper bound on sums of reciprocals of products of fractional parts and sheds some light on a recent question of Lê and Vaaler.  相似文献   

3.
For \(0<\alpha <\infty \), \(0<p<\infty \) and \(0<s<\infty \), we characterize the closures in the \(\alpha \)-Bloch norm of \(\alpha \)-Bloch functions that are in a Hardy space \(H^p\) and in a Hardy–Sobolev space \(H^p_s\) on the unit ball of \(\mathbb {C}^n\).  相似文献   

4.
Let k be an integer with \(k\ge 3\) and \(\eta \) be any real number. Suppose that \(\lambda _1, \lambda _2, \lambda _3, \lambda _4, \mu \) are non-zero real numbers, not all of the same sign and \(\lambda _1/\lambda _2\) is irrational. It is proved that the inequality \(|\lambda _1p_1^2+\lambda _2p_2^2+\lambda _3p_3^2+\lambda _4p_4^2+\mu p_5^k+\eta |<(\max \ p_j)^{-\sigma }\) has infinitely many solutions in prime variables \(p_1, p_2, \ldots , p_5\), where \(0<\sigma <\frac{1}{16}\) for \(k=3,\ 0<\sigma <\frac{5}{3k2^k}\) for \(4\le k\le 5\) and \(0<\sigma <\frac{40}{21k2^k}\) for \(k\ge 6\). This gives an improvement of an earlier result.  相似文献   

5.
A stable-like Markov chain is a time-homogeneous Markov chain on the real line with the transition kernel \(p(x,\hbox {d}y)=f_x(y-x)\hbox {d}y\), where the density functions \(f_x(y)\), for large \(|y|\), have a power-law decay with exponent \(\alpha (x)+1\), where \(\alpha (x)\in (0,2)\). In this paper, under a certain uniformity condition on the density functions \(f_x(y)\) and additional mild drift conditions, we give sufficient conditions for recurrence in the case when \(0<\liminf _{|x|\longrightarrow \infty }\alpha (x)\), sufficient conditions for transience in the case when \(\limsup _{|x|\longrightarrow \infty }\alpha (x)<2\) and sufficient conditions for ergodicity in the case when \(0<\inf \{\alpha (x):x\in \mathbb {R}\}\). As a special case of these results, we give a new proof for the recurrence and transience property of a symmetric \(\alpha \)-stable random walk on \(\mathbb {R}\) with the index of stability \(\alpha \ne 1\).  相似文献   

6.
We consider the remaining unsettled cases in the problem of existence of energy minimizing solutions for the Dirichlet value problem \(L_\gamma u-\lambda u=\frac{u^{2^*(s)-1}}{|x|^s}\) on a smooth bounded domain \(\Omega \) in \({\mathbb {R}}^n\) (\(n\ge 3\)) having the singularity 0 in its interior. Here \(\gamma <\frac{(n-2)^2}{4}\), \(0\le s <2\), \(2^*(s):=\frac{2(n-s)}{n-2}\) and \(0\le \lambda <\lambda _1(L_\gamma )\), the latter being the first eigenvalue of the Hardy–Schrödinger operator \(L_\gamma :=-\Delta -\frac{\gamma }{|x|^2}\). There is a threshold \(\lambda ^*(\gamma , \Omega ) \ge 0\) beyond which the minimal energy is achieved, but below which, it is not. It is well known that \(\lambda ^*(\Omega )=0\) in higher dimensions, for example if \(0\le \gamma \le \frac{(n-2)^2}{4}-1\). Our main objective in this paper is to show that this threshold is strictly positive in “lower dimensions” such as when \( \frac{(n-2)^2}{4}-1<\gamma <\frac{(n-2)^2}{4}\), to identify the critical dimensions (i.e., when the situation changes), and to characterize it in terms of \(\Omega \) and \(\gamma \). If either \(s>0\) or if \(\gamma > 0\), i.e., in the truly singular case, we show that in low dimensions, a solution is guaranteed by the positivity of the “Hardy-singular internal mass” of \(\Omega \), a notion that we introduce herein. On the other hand, and just like the case when \(\gamma =s=0\) studied by Brezis and Nirenberg (Commun Pure Appl Math 36:437–477, 1983) and completed by Druet (Ann Inst H Poincaré Anal Non Linéaire 19(2):125–142, 2002), \(n=3\) is the critical dimension, and the classical positive mass theorem is sufficient for the merely singular case, that is when \(s=0\), \(\gamma \le 0\).  相似文献   

7.
We study transience and recurrence of simple random walks on percolation clusters in the hierarchical group of order N, which is an ultrametric space. The connection probability on the hierarchical group for two points separated by distance k is of the form \(c_k/N^{k(1+\delta )}, \delta >0\), with \(c_k=C_0+C_1\log k+C_2k^\alpha \), non-negative constants \(C_0, C_1, C_2\), and \(\alpha >0\). Percolation occurs for \(\delta <1\), and for the critical case, \(\delta =1\), \(\alpha >0\) and sufficiently large \(C_2\). We show that in the case \(\delta <1\) the walk is transient, and in the case \(\delta =1,C_2>0,\alpha >0\) there exists a critical \(\alpha _\mathrm{c}\in (0,\infty )\) such that the walk is recurrent for \(\alpha <\alpha _\mathrm{c}\) and transient for \(\alpha >\alpha _\mathrm{c}\). The proofs involve ultrametric random graphs, graph diameters, path lengths, and electric circuit theory. Some comparisons are made with behaviours of simple random walks on long-range percolation clusters in the one-dimensional Euclidean lattice.  相似文献   

8.
We establish upper bounds for the number of primitive integer solutions to inequalities of the shape \(0<|F(x, y)| \le h\), where \(F(x , y) =(\alpha x + \beta y)^r -(\gamma x + \delta y)^r \in \mathbb {Z}[x ,y]\), \(\alpha \), \(\beta \), \(\gamma \) and \(\delta \) are algebraic constants with \(\alpha \delta -\beta \gamma \ne 0\), and \(r \ge 5\) and h are integers. As an important application, we pay special attention to binomial Thue’s inequalities \(|ax^r - by^r| \le c\). The proofs are based on the hypergeometric method of Thue and Siegel and its refinement by Evertse.  相似文献   

9.
Let \(n\in \mathbb {N}\), \(n\ge 2\), \(\beta >0\) fixed, and \(0<b\le \beta \). For \(n-1<\alpha \le n\), we look to classify extremal points for the fractional differential equation \(D_{0^+}^{\alpha }u+p(t) u=0\), satisfying the boundary conditions \(u^{(i)}(0)=0\), \(i=0,\ldots ,n-2\), \(D_{0^+}^\gamma u(b)=0\), where p(t) is a continuous nonnegative function on \([0,\beta ]\) which does not vanish identically on any nondegenerate compact subinterval of \([0,\beta ]\). Using the theory of Krein and Rutman, first extremal points of this boundary value problem are classified. As an application, the results are applied, along with a fixed-point theorem, to show the existence of a solution of a nonlinear fractional boundary value problem.  相似文献   

10.
We prove some sharp \(L^p-L^2\) estimates for joint spectral projections \(\pi _{\ell \ell '}\), with \(\ell ,\ell '\in {\mathbb {N}}\), \(\ell \ge \ell '\ge 0\), \(1\le p\le 2\), associated to the Laplace–Beltrami operator and to a suitably defined subLaplacian on the unit quaternionic sphere.  相似文献   

11.
We consider the following fractional \( p \& q\) Laplacian problem with critical Sobolev–Hardy exponents
$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}_{p} u + (-\Delta )^{s}_{q} u = \frac{|u|^{p^{*}_{s}(\alpha )-2}u}{|x|^{\alpha }}+ \lambda f(x, u) &{} \text{ in } \Omega \\ u=0 &{} \text{ in } \mathbb {R}^{N}{\setminus } \Omega , \end{array} \right. \end{aligned}$$
where \(0<s<1\), \(1\le q<p<\frac{N}{s}\), \((-\Delta )^{s}_{r}\), with \(r\in \{p,q\}\), is the fractional r-Laplacian operator, \(\lambda \) is a positive parameter, \(\Omega \subset \mathbb {R}^{N}\) is an open bounded domain with smooth boundary, \(0\le \alpha <sp\), and \(p^{*}_{s}(\alpha )=\frac{p(N-\alpha )}{N-sp}\) is the so-called Hardy–Sobolev critical exponent. Using concentration-compactness principle and the mountain pass lemma due to Kajikiya [23], we show the existence of infinitely many solutions which tend to be zero provided that \(\lambda \) belongs to a suitable range.
  相似文献   

12.
In this paper, by using the \(\alpha \)-resolvent family theory, Banach contraction mapping principle and Schauder’s fixed point theorem, we investigate the existence of anti-periodic mild solutions to the semilinear fractional differential equations \(D^{\alpha }_{t}u(t) = Au(t) +f(t,u(t)),\ t\in R,1 \le \alpha \le 2 \) and \(D^{\alpha }_{t}u(t) = Au(t) +f(t,u(t),u'(t)),\ t\in R,1 < \alpha < 2\), where \(A : D(A)\subset X \rightarrow X\) is the infinitesimal generator of an \(\alpha \)-resolvent family defined on a Banach space \(X\) and \(f\) is a suitable function. Furthermore, an example is given to illustrate our results.  相似文献   

13.
We consider random matrices of the form \(H = W + \lambda V, \lambda \in {\mathbb {R}}^+\), where \(W\) is a real symmetric or complex Hermitian Wigner matrix of size \(N\) and \(V\) is a real bounded diagonal random matrix of size \(N\) with i.i.d. entries that are independent of \(W\). We assume subexponential decay of the distribution of the matrix entries of \(W\) and we choose \(\lambda \sim 1\), so that the eigenvalues of \(W\) and \(\lambda V\) are typically of the same order. Further, we assume that the density of the entries of \(V\) is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is \(\lambda _+\in {\mathbb {R}}^+\) such that the largest eigenvalues of \(H\) are in the limit of large \(N\) determined by the order statistics of \(V\) for \(\lambda >\lambda _+\). In particular, the largest eigenvalue of \(H\) has a Weibull distribution in the limit \(N\rightarrow \infty \) if \(\lambda >\lambda _+\). Moreover, for \(N\) sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for \(\lambda >\lambda _+\), while they are completely delocalized for \(\lambda <\lambda _+\). Similar results hold for the lowest eigenvalues.  相似文献   

14.
This paper is concerned with the existence of positive solution to a class of singular fourth order elliptic equation of Kirchhoff type
$$\begin{aligned} \triangle ^2 u-\lambda M(\Vert \nabla u\Vert ^2)\triangle u-\frac{\mu }{\vert x\vert ^4}u=\frac{h(x)}{u^\gamma }+k(x)u^\alpha , \end{aligned}$$
under Navier boundary conditions, \(u=\triangle u=0\). Here \(\varOmega \subset {\mathbf {R}}^N\), \(N\ge 1\) is a bounded \(C^4\)-domain, \(0\in \varOmega \), h(x) and k(x) are positive continuous functions, \(\gamma \in (0,1)\), \(\alpha \in (0,1)\) and \(M:{\mathbf {R}}^+\rightarrow {\mathbf {R}}^+\) is a continuous function. By using Galerkin method and sharp angle lemma, we will show that this problem has a positive solution for \(\lambda > \frac{\mu }{\mu ^*m_0}\) and \(0<\mu <\mu ^*\). Here \(\mu ^*=\Big (\frac{N(N-4)}{4}\Big )^2\) is the best constant in the Hardy inequality. Besides, if \(\mu =0\), \(\lambda >0\) and hk are Lipschitz functions, we show that this problem has a positive smooth solution. If \(h,k\in C^{2,\,\theta _0}(\overline{\varOmega })\) for some \(\theta _0\in (0,1)\), then this problem has a positive classical solution.
  相似文献   

15.
Let \(\kappa \) be an infinite cardinal, and \(2^\kappa <\lambda \le 2^{\kappa ^+}\). We prove that if there is a weak diamond on \(\kappa ^+\) then every \(\{C_\alpha :\alpha <\lambda \}\subseteq \mathcal {D}_{\kappa ^+}\) satisfies Galvin’s property. On the other hand, Galvin’s property is consistent with the failure of the weak diamond (and even with Martin’s axiom in the case of \(\aleph _1\)). We derive some consequences about weakly inaccessible cardinals. We also prove that the negation of a similar property follows from the proper forcing axiom.  相似文献   

16.
A bounded linear operator T acting on a Hilbert space is said to have orthogonality property \(\mathcal {O}\) if the subspaces \(\ker (T-\alpha )\) and \(\ker (T-\beta )\) are orthogonal for all \(\alpha , \beta \in \sigma _p(T)\) with \(\alpha \ne \beta \). In this paper, the authors investigate the compact perturbations of operators with orthogonality property \(\mathcal {O}\). We give a sufficient and necessary condition to determine when an operator T has the following property: for each \(\varepsilon >0\), there exists \(K\in \mathcal {K(H)}\) with \(\Vert K\Vert <\varepsilon \) such that \(T+K\) has orthogonality property \(\mathcal {O}\). Also, we study the stability of orthogonality property \(\mathcal {O}\) under small compact perturbations and analytic functional calculus.  相似文献   

17.
In this note we improve the standard regularity of the dynamic part of the pressure in the Navier–Stokes system. Using the theory of elliptic equations with \(L^1\) right-hand side we prove that, in addition to be in \(L^2\), the dynamic pressure belongs to \(W^{1,\alpha }_{loc} \) with \(1<\alpha <\frac{n}{n-1}\), in case of Dirichlet boundary condition. For pressure boundary condition the dynamic pressure is proved to be in \(W^{1,\alpha } \). As a consequence, for the force \(\mathbf{f} \in L^q (\Omega )^n \) and \(q>n /2 \) the pressure turns out to be continuous.  相似文献   

18.
We study the asymptotic tail behaviour of the first passage time over a moving boundary for asymptotically \(\alpha \)-stable Lévy processes with \(\alpha <1\). Our main result states that if the left tail of the Lévy measure is regularly varying with index \(- \alpha \), and the moving boundary is equal to \(1 - t^{\gamma }\) for some \(\gamma <1/\alpha \), then the probability that the process stays below the moving boundary has the same asymptotic polynomial order as in the case of a constant boundary. The same is true for the increasing boundary \(1 + t^{\gamma }\) with \(\gamma <1/\alpha \) under the assumption of a regularly varying right tail with index \(-\alpha \).  相似文献   

19.
Let \(\{X(t):t\in \mathbb R_+\}\) be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, \(\mathbb E X(t) = 0, \mathbb E X^2(t) = 1\) and correlation function satisfying (i) \(r(t) = 1 - C|t|^{\alpha } + o(|t|^{\alpha })\) as \(t\rightarrow 0\) for some \(0\le \alpha \le 2\) and \(C>0\); (ii) \(\sup _{t\ge s}|r(t)|<1\) for each \(s>0\) and (iii) \(r(t) = O(t^{-\lambda })\) as \(t\rightarrow \infty \) for some \(\lambda >0\). For any \(n\ge 1\), consider n mutually independent copies of X and denote by \(\{X_{r:n}(t):t\ge 0\}\) the rth smallest order statistics process, \(1\le r\le n\). We provide a tractable criterion for assessing whether, for any positive, non-decreasing function \(f, \mathbb P(\mathscr {E}_f)=\mathbb P(X_{r:n}(t) > f(t)\, \text { i.o.})\) equals 0 or 1. Using this criterion we find, for a family of functions \(f_p(t)\) such that \(z_p(t)=\mathbb P(\sup _{s\in [0,1]}X_{r:n}(s)>f_p(t))=O((t\log ^{1-p} t)^{-1})\), that \(\mathbb P(\mathscr {E}_{f_p})= 1_{\{p\ge 0\}}\). Consequently, with \(\xi _p (t) = \sup \{s:0\le s\le t, X_{r:n}(s)\ge f_p(s)\}\), for \(p\ge 0\) we have \(\lim _{t\rightarrow \infty }\xi _p(t)=\infty \) and \(\limsup _{t\rightarrow \infty }(\xi _p(t)-t)=0\) a.s. Complementarily, we prove an Erdös–Révész type law of the iterated logarithm lower bound on \(\xi _p(t)\), namely, that \(\liminf _{t\rightarrow \infty }(\xi _p(t)-t)/h_p(t) = -1\) a.s. for \(p>1\) and \(\liminf _{t\rightarrow \infty }\log (\xi _p(t)/t)/(h_p(t)/t) = -1\) a.s. for \(p\in (0,1]\), where \(h_p(t)=(1/z_p(t))p\log \log t\).  相似文献   

20.
We consider the positive solutions of the nonlinear eigenvalue problem \(-\Delta _{\mathbb {H}^n} u = \lambda u + u^p, \) with \(p=\frac{n+2}{n-2}\) and \(u \in H_0^1(\Omega ),\) where \(\Omega \) is a geodesic ball of radius \(\theta _1\) on \(\mathbb {H}^n.\) For radial solutions, this equation can be written as an ordinary differential equation having n as a parameter. In this setting, the problem can be extended to consider real values of n. We show that if \(2<n<4\) this problem has a unique positive solution if and only if \(\lambda \in \left( n(n-2)/4 +L^*\,,\, \lambda _1\right) .\) Here \(L^*\) is the first positive value of \(L = -\ell (\ell +1)\) for which a suitably defined associated Legendre function \(P_{\ell }^{-\alpha }(\cosh \theta ) >0\) if \(0 < \theta <\theta _1\) and \(P_{\ell }^{-\alpha }(\cosh \theta _1)=0,\) with \(\alpha = (2-n)/2\).  相似文献   

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